The team looks over the problem statement for the new process improvement project.

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One of the team members says, “What exactly are we trying lớn solve? I know that we are trying lớn get out more products over a shift, but I have no idea as lớn what we are trying to vì chưng to make that possible.”

“I’m still clueless regarding how we plan lớn figure this out. What are we even looking at changing? Are we changing where work is done, the tooling involved, or possibly the personnel? What factors influence the throughput rate, & how do we go about proving that our solution was the right solution?”

The project manager is pleased that the team is asking the right questions. It’s not enough to lớn just see if the throughput rate improved, but it is important to understand how a change in the system inputs affect the throughput rate, or results of the process. In the world of Six Sigma, this concept is known as Y=f(x).

An overview: What is Y=f(x)?

In the strictest view, Y=f(x) is a representation of a mathematical formula. It is one to lớn use when examining different possible outcomes based on the inputs & factors used. The “Y” stands for the outcome, the “f” embodies the function used in the calculation, & the “X” represents the input or inputs used for the formula. 

This formula, when associated with Six Sigma, is called the breakthrough equation. So, we use a formula lớn find out which inputs will send us khổng lồ the optimal, or best, output.

The formula is used throughout a problem-solving process. For example, when using the DMAIC steps, we progress through the following five stages:

Define: Gather the desired outcome (Y)Measure: Gather the possible inputs (x) as well as measurements for inputs (x) và outputs (Y)Analyze: Use the formula (f) to kiểm tra the relationship between inputs (x) và outcome (Y)Improve: Select and implements the best inputs (x) for achieving the desired outcome (Y)Control: Monitor the Y over time khổng lồ see if any changes in inputs (x) are occurring

The benefits of attending khổng lồ Y=f(x) 

Assists in expressing the desired outcome as a measurable term

It can be different setting up the initial problem statement, và the various “who, what, when, where, why, và how many” questions. A key element of the problem statement is determining the metric we’re trying to improve. 

When using Y=f(x) in the Define stage, it pushes us to clearly understand “Y,” which is the desired results we are working to achieve. By using the formula setup, it forces the team to lớn view the outcome as a measurable term.

Guides the results for process improvement efforts

It can be easy to thua trận track of how we measure process improvement goals during a process improvement project. Using Y=f(x) as a concept forces the project manager lớn consider metrics from the beginning of the project all the way lớn the over of the project. These projects need to lớn show quantitative improvement, so understanding the desired outcome và how inputs affect reaching our project goals is critical.

Makes it possible lớn monitor improvements over time

By understanding the inputs & outputs of a continuous improvement project, it becomes possible lớn understand how lớn verify the results once implemented, và the best way to lớn monitor that the results are maintained over time.

Best practices for using Y=f(x)

By setting up the formula, it assists us in selecting the right tool lớn verify X-Y relationships.Be sure that as many inputs that are significant factors connected khổng lồ the results are included. Consider statistical processing software, lượt thích Minitab, for examining whether the combination of specific inputs causes a greater (or lesser) impact on the results.Remember to use Y=f(x) at all stages of problem-solving. Forget to lớn use it at the beginning, và you may be working on the wrong problem, or with the incorrect formula. Forget to use it at the end, and you may find out that either the anticipated savings were not met, or that they were not maintained. 

Frequently Asked Questions (FAQ) about Y=f(x)

I have seen a version of Y=f(x) that included a “+E” component at the kết thúc of the equation. Is this a different concept?

No, this is still the Y=f(x) methodology, but it adds an additional mathematical unique into play. The “+E” stands for an element of random error related lớn the input “x” when transformed by the function “f.”

Does Y=f(x) methodology only work with DMAIC?

While the example above matches up how Y=f(x) is used during the various stages of the DMAIC problem solving process, Y=f(x) is a concept that also matches up with the stages of other problem-solving methodologies.

Do I need khổng lồ monitor every x and Y from the project once I enter the control phase of DMAIC?

While there is nothing stopping you from doing so, it’s recommended khổng lồ use the lessons learned from the event and only monitor the most important inputs (x) per the conclusions of the project. 

It’s common for projects to fail at the control phase, so the easier you can make monitoring the metrics, the more likely the team is to lớn successfully perform the monitoring steps.

Y=f(x) guides the team to understand how inputs affect the results, leading to better solutions

Y=f(x) is a concept within Six Sigma and other problem-solving methodologies that connects two concepts: Results should be measurable, & there should be an understanding between how inputs affect the results. Problem-solving should involve the attempt to find the best set of inputs that help drive a process to the optimal result.

Because Six Sigma approaches things with a statistical mindset, it considers all problems as a function. Using mathematical symbols, this looks like:


The y=f(x) statement can be used in two ways. First, it is a general bản đồ for stating a problem. Y (the problem) occurs because some X (input or cause) is occurring. In reality, Y is usually occurring because of some group of causes or inputs, which means there are going lớn be more than one X inputs.

The idea can also be applied to lớn specific processes and outcomes within the problem. As you get more & more granular, the y=f(x) concept becomes increasingly mathematical; in many cases, you can graph the relationship between the output (y) & the input (x).

To understand the concept of thinking of problems as a function, let’s look at a problem that might occur for a large internet service provider. The manager of a service team has discovered that service calls are taking much longer than expected; in fact, his five team members take 1.75 times longer on average than other service reps in the company khổng lồ handle all types of calls.

To find out what might be causing the situation, the manager researches the problem by talking khổng lồ the reps, talking lớn the customers, & going out on random calls with all five representatives. He makes the following observations:

One representative is a native khổng lồ the area the team services, which means he or she knows many of the customers personally. This results in friendly chatter that lengthens the time on the job.One representative is providing customers with very in-depth explanations & education about internet issues, sometimes over và beyond what the customers would ever need to know regarding their mạng internet service.One representative is new to lớn the job & takes longer to complete each task because he or she is unsure of the work, has khổng lồ double-check the work, or calls another rep lớn ask questions about the work.The remaining two reps perform work in times that are on par with company averages.

The manager distills this data down to two overall causes for the problem:

Too much talking (reps one and two)Inadequate training

The problem can now be stated as a function:

The extra time is a function of too much talking và inappropriate training.

The manager also now has two root causes to address. The example is simple, but it illustrates the basic concept in defining a y=f(x) relationship for a problem and its causes. It’s not always so easy to lớn conduct the research & analysis to lớn find the relationship, but the relationship is always present.

Some other examples of y=f(x) relationships include:

Low customer satisfaction with hamburger taste is a function of an uncalibrated grill.Low employee morale is a function of a poor time-off approval system.Customer wait times are a function of technology distractions for employees.

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